In this Letter, we turn to the investigation of asymptotically AdS
spacetimes generated by static and spinning string sources in
four-dimensional Einstein-Maxwell-dilaton theory which are
horinzonless and have nontrivial external solutions. The
motivation for studying such kinds of solutions is that they may
be interpreted as cosmic strings. Cosmic strings are topological
structure that arise from the possible phase transitions to which
the universe might have been subjected to and may play an
important role in the formation of primordial structures. A short
review of papers treating this subject follows. The
four-dimensional horizonless solutions of Einstein gravity have
been explored in [1]}, [2]}. These horizonless solutions
[1]}, [2]} have a conical geometry; they are everywhere flat
except at the location of the line source. The spacetime can be
obtained from the flat spacetime by cutting out a wedge and
identifying its edges. The wedge has an opening angle which turns
to be proportional to the source mass. The extension to include
the Maxwell field has also been done [5]}. Static and
spinning magnetic sources in three and four-dimensional
Einstein-Maxwell gravity with negative cosmological constant have
been explored in [6]}, [7]}. The generalization of these
asymptotically AdS magnetic rotating solutions to higher
dimensions has also been done [8]}. In the context of
electromagnetic cosmic string, it has been shown that there are
cosmic strings, known as superconducting cosmic strings, that
behave as superconductors and have interesting interactions with
astrophysical magnetic fields [9]}. The properties of these
superconducting cosmic strings have been investigated in
[10]}. It is also of great interest to generalize the study
to the dilaton gravity theory [11]}. While exact magnetic
rotating dilaton solution in three dimensions has been obtained in
[12]}, two classes of magnetic rotating solutions in four
[13]} and higher dimensional dilaton gravity in the presence
of one Liouville-type potential have been constructed [14]}.
Unfortunately, these solutions [13]}, [14]} are neither
asymptotically flat nor (A)dS. The purpose of the present Letter
is to construct a new class of static and spinning magnetic
dilaton string solutions which produces a longitudinal magnetic
field in the background of anti-de Sitter spacetime. We will also
present the suitable counterterm which removes the divergences of
the action, and calculate the conserved quantities by using the
counterterm method.
